CoCalc Public Fileswww / talks / harvard-talk-2004-12-08 / e2heights / mazur_request.txt
Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
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3Karl Rubin and I were wondering whether you could do a p-adic height
4computation, the point of which is to figure out whether the p-adic
5regulator of elliptic curves has a "tendency" to be not divisible by p. We
6want to avoid a few degenerate cases, so the smallest p that is useful to
7us is p=5. Is the strategy-- described below--, that systematically runs
9(non-3-Eisenstein) elliptic curve of conductor 37 reasonably easy to do?
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13Our untwisted equation is E:  y^2 = 4x^3-4x+1. For any nonzero integer D we
14consider the twisted elliptic curve  E_D:  D.y^2 = 4x^3-4x+1.  Put F(x): =
154x^3-4x+1. Let x run through integers
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17  a=  0, \pm 1, \pm 2, \pm3, ...
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19and for each of these a's,
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211) Factor F(a) = d.b^2,  where d is square-free;
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232) throw out the ones where d is divisible by 5, or where the parity of the
24rank of E_d is even;
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263) for the rest, viewing P:=(a,b) as a rational point-- it will be of
27infinite order--on the elliptic curve E_d, so one can compute the 5-adic
28height of P on the elliptic curve E_d: this should be a 5-adic integer if
29the height is normalized right, and there should be a good number of
30instances where  this height is a 5-adic unit; throw them out, and
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324) print the rest.
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35Note that we only want to know the 5-adic height modulo 5.
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37Then it would be our job to figure out whether strange things occur for
38this printed list of instances: e.g., the point P might be divisible by 5,
39or the Mordell-Weil rank might be > 2, or other stuff..., and we want to
40know what other stuff there might be.
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